Abstract
A recently developed high-frequency asymptotic solution for the famous “Sommerfeld radiation problem” is revisited. The solution is based on an analysis performed in the spectral domain, through which a compact asymptotic formula describes the behavior of the EM field, which emanates from a vertical Hertzian radiating dipole, located above flat, lossy ground. The paper is divided into two parts. We first demonstrate an efficient technique for the accurate numerical calculation of the well-known Sommerfeld integrals. The results are compared against alternative calculation approaches and validated with the corresponding Norton figures for the surface wave. In the second part, we introduce the asymptotic solution and investigate its performance; we compare the solution with the accurate numerical evaluation for the received EM field and with a more basic asymptotic solution to the given problem, obtained via the application of the Stationary Phase Method. Simulations for various frequencies, distances, altitudes, and ground characteristics are illustrated and inferences for the applicability of the solution are made. Finally, special cases leading to analytical field expressions close as well as far from the interface are examined.
Highlights
We show that, using an appropriate variable transformation, it is possible to convert the generalized integrals of [23] into fast converging formulas, which are rather suitable for numerical calculation, using standard Numerical Integration (NI) techniques
Pay attention to the fact that this is not a true surface wave, at least when one of the accepted definitions for a type of surface wave is considered [45]
We demonstrated an efficient method for the numerical calculation of Sommerfeld integrals
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The original Sommerfeld solution is provided in the spatial domain as an integral expression, utilizing the so-called “Hertz potentials”, but it does not end up in analytic formulas [1]. Sommerfeld further derived an asymptotic surface wave expression, assuming high media contrast and large horizontal range. A. Norton focused on the engineering application of the problem; provided approximate solutions, represented by long algebraic expressions; and described concepts such as the propagating surface wave and its associated “attenuation coefficient”, albeit his definition for the surface wave is not equivalent to Sommerfeld’s definition [6,7]. √ that a slowly decaying EM field near the interface that decreases approximately as of 1/ R can be excited and detected under usual circumstances, providing a tangible indication as for the existence of Zenneck waves [9,10]. A thorough review of the subject is given in [19,20]
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